Oscillatory properties of the second order nonlinear equation \[ (r(t)x^{\prime })^{\prime }+q(t)f(x)=0 \] are investigated. In particular, criteria for the existence of at least one oscillatory solution and for the global monotonicity properties of nonoscillatory solutions are established. The possible coexistence of oscillatory and nonoscillatory solutions is studied too.
For the equation y (n) + |y| k sgn y = 0, k > 1, n = 3, 4, existence of oscillatory solutions y = (x ∗ − x) −α h(log(x ∗ − x)), α = n ⁄ k − 1 , x < x∗ , is proved, where x ∗ is an arbitrary point and h is a periodic non-constant function on R. The result on existence of such solutions with a positive periodic non-constant function h on R is formulated for the equation y (n) = |y| k sgn y, k > 1, n = 12, 13, 14.
The paper deals with the oscillation of a differential equation $L_4y+P(t)L_2y+Q(t)y\equiv 0$ as well as with the structure of its fundamental system of solutions.
Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation y (n) + ∑ n−1 j=0 aj (x)y (j) + p(x)|y| k sgn y = 0 with n > 1, real (not necessarily natural) k > 1, and continuous functions p(x) and aj (x) defined in a neighborhood of +∞. For this equation with positive potential p(x) a criterion is formulated for existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. Sufficient conditions are obtained for existence of solution to this equation which is equivalent to a polynomial.
Consider the difference equation ∆x(n) +∑m i=1 pi(n)x(τi(n)) = 0, n ≥ 0 [ ∇x(n) − ∑m i=1 pi(n)x(σi(n)) = 0, n ≥ 1 ] , where (pi(n)), 1 6 i 6 m are sequences of nonnegative real numbers, τi(n) [σi(n)], 1 6 i 6 m are general retarded (advanced) arguments and ∆ [∇] denotes the forward (backward) difference operator ∆x(n) = x(n + 1) − x(n) [∇x(n) = x(n) − x(n − 1)]. New oscillation criteria are established when the well-known oscillation conditions lim sup n→∞ ∑m i=1 ∑n j=τ(n) pi(j) > 1 [ lim sup n→∞ ∑m i=1 σ∑ (n) j=n pi(j) > 1 ] and lim inf n→∞ ∑m i=1 n∑−1 j=τi(n) pi(j) > 1⁄e [ lim inf n→∞ ∑m i=1 σ∑i(n) j=n+1 pi(j) > 1⁄e ] are not satisfied. Here τ (n) = max 1≤i≤m τi(n) [σ(n) = min 1≤i≤m σi(n)]. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.
We obtain sufficient conditions for every solution of the differential equation [y(t) − p(t)y(r(t))](n) + v(t)G(y(g(t))) − u(t)H(y(h(t))) = f(t) to oscillate or to tend to zero as t approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when G has sub-linear growth at infinity. Our results also apply to the neutral equation [y(t) − p(t)y(r(t))](n) + q(t)G(y(g(t))) = f(t) when q(t) has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.